Transformation of coordinates

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 33 | 1 | 2 |

Section 1.3.2.3.9.5. Transformation of coordinates

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.3.9.5. Transformation of coordinates

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Let σ be a smooth non-singular change of variables in $[{\bb R}^{n}]$ , i.e. an infinitely differentiable mapping from an open subset Ω of $[{\bb R}^{n}]$ to Ω′ in $[{\bb R}^{n}]$ , whose Jacobian $[J(\sigma) = \det \left[{\partial \sigma ({\bf x}) \over \partial {\bf x}}\right]]$ vanishes nowhere in Ω. By the implicit function theorem, the inverse mapping $[\sigma^{-1}]$ from Ω′ to Ω is well defined.

If f is a locally summable function on Ω, then the function $[\sigma^{\#} f]$ defined by $[(\sigma^{\#} f)({\bf x}) = f[\sigma^{-1}({\bf x})]]$ is a locally summable function on Ω′, and for any $[\varphi \in {\scr D}(\Omega')]$ we may write: $[\eqalign{{\textstyle\int\limits_{\Omega'}} (\sigma^{\#} f) ({\bf x}) \varphi ({\bf x}) \;\hbox{d}^{n} {\bf x} &= {\textstyle\int\limits_{\Omega'}} f[\sigma^{-1} ({\bf x})] \varphi ({\bf x}) \;\hbox{d}^{n} {\bf x} \cr &= {\textstyle\int\limits_{\Omega'}} f({\bf y}) \varphi [\sigma ({\bf y})]|J(\sigma)| \;\hbox{d}^{n} {\bf y} \quad \hbox{by } {\bf x} = \sigma ({\bf y}).}]$ In terms of the associated distributions $[\langle T_{\sigma^{\#} f}, \varphi \rangle = \langle T_{f}, |J(\sigma)|(\sigma^{-1})^{\#} \varphi \rangle.]$

This operation can be extended to an arbitrary distribution T by defining its image $[\sigma^{\#} T]$ under coordinate transformation σ through $[\langle \sigma^{\#} T, \varphi \rangle = \langle T, |J(\sigma)|(\sigma^{-1})^{\#} \varphi \rangle,]$ which is well defined provided that σ is proper, i.e. that $[\sigma^{-1}(K)]$ is compact whenever K is compact.

For instance, if $[\sigma: {\bf x} \;\longmapsto\; {\bf x} + {\bf a}]$ is a translation by a vector a in $[{\bb R}^{n}]$ , then $[|J(\sigma)| = 1]$ ; $[\sigma^{\#}]$ is denoted by $[\tau_{\bf a}]$ , and the translate $[\tau_{\bf a} T]$ of a distribution T is defined by $[\langle \tau_{\bf a} T, \varphi \rangle = \langle T, \tau_{-{\bf a}} \varphi \rangle.]$

Let $[A: {\bf x} \;\longmapsto\; {\bf Ax}]$ be a linear transformation defined by a non-singular matrix A. Then $[J(A) = \det {\bf A}]$ , and $[\langle A^{\#} T, \varphi \rangle = |\det {\bf A}| \langle T, (A^{-1})^{\#} \varphi \rangle.]$ This formula will be shown later (Sections 1.3.2.6.5, 1.3.4.2.1.1) to be the basis for the definition of the reciprocal lattice.

In particular, if $[{\bf A} = -{\bf I}]$ , where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting $[A^{\#} \varphi]$ by $[\breve{\varphi}]$ we have: $[\langle \breve{T}, \varphi \rangle = \langle T, \breve{\varphi} \rangle.]$ T is called an even distribution if $[\breve{T} = T]$ , an odd distribution if $[\breve{T} = -T]$ .

If $[{\bf A} = \lambda {\bf I}]$ with $[\lambda \gt 0]$ , A is called a dilation and $[\langle A^{\#} T, \varphi \rangle = \lambda^{n} \langle T, (A^{-1})^{\#} \varphi \rangle.]$ Writing symbolically δ as $[\delta ({\bf x})]$ and $[A^{\#} \delta]$ as $[\delta ({\bf x}/\lambda)]$ , we have: $[\delta ({\bf x}/\lambda) = \lambda^{n} \delta ({\bf x}).]$ If [n = 1] and f is a function with isolated simple zeros $[x_{j}]$ , then in the same symbolic notation $[\delta [\;f(x)] = \sum\limits_{j} {1 \over |\;f'(x_{j})|} \delta (x_{j}),]$ where each $[\lambda_{j} = 1/|\;f'(x_{j})|]$ is analogous to a `Lorentz factor' at zero $[x_{j}]$ .

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 33